Robust Reentry Guidance of a Reusable Launch Vehicle
using Model Predictive Static Programming
Omkar Halbe1 , Ramsingh G. Raja2 and Radhakant Padhi3
Indian Institute of Science, Bangalore, INDIA
A robust suboptimal reentry guidance scheme is presented for a reusable launch vehicle using the
recently-developed computationally efficient model predictive static programming. The formulation
uses the nonlinear vehicle dynamics with spherical and rotating earth, hard constraints for desired terminal conditions and an innovative cost function having several components with associated weighting
factors that can account for path and control constraints in a soft constraint manner, thereby leading
to smooth solutions of the guidance parameters. The proposed guidance essentially shapes the trajectory of the vehicle by computing the necessary angle of attack and bank angle that the vehicle should
execute. The path constraints are the structural load constraint, thermal load constraint, bounds on
the angle of attack and bounds on the bank angle. In addition, the terminal constraints are the threedimensional position and velocity vector components at the end of the reentry. Where as the angle of
attack command is generated directly, the bank angle command is generated by first generating the
required heading angle history and then using it in a dynamic inversion loop considering the heading
angle dynamics. Such a two-loop synthesis of bank angle leads to better management of the vehicle
trajectory and avoids mathematical complexity as well. Moreover, all bank angle maneuvers have been
confined to the middle of the trajectory and the vehicle ends the reentry segment with near zero bank
angle, which is quite desirable. It has also been demonstrated that the proposed guidance has sufficient
robustness for state perturbations as well as parametric uncertainties in the model.
1
2
3
Former Project Associate, Department of Aerospace Engineering
Project Associate, Department of Aerospace Engineering
Associate Professor (Associate Fellow, AIAA), Dept. of Aerospace Engineering. padhi@aero.iisc.ernet.in.
An earlier version of this paper was presented in 2010 AIAA GNC conference (Paper No. AIAA-2010-8311).
1
NOMENCLATURE
α
Angle of attack, rad
σ
Bank angle, rad
h
Height from earth’s surface, m
r
Radial distance from earth’s center, m
V
Velocity, m/sec
VCO
Circular orbital velocity, m/sec
γ
Flight path angle, rad
φ
Latitude angle, rad
θ
Longitude angle, rad
ψ
Heading angle, rad
t
Time, sec
e
Energy height, m
Ωe
Earth rotational velocity, rad/sec
M
Mach number
ρ
Atmospheric density, kg/m3
ρSL
Atmospheric density at sea level, kg/m3
m
Mass of vehicle, kg
RN
Nose radius of vehicle, m
q
Dynamic pressure, N/m2
L
Lift, N
D
Drag, N
g
Gravity at earth’s surface, m/sec2
Sre f
Reference surface area, m
Re
Earth radius, m
Nz
Normal load, g
CL , CD Lift and drag force coefficients respectively
2
Acronyms
RLV
Reusable Launch Vehicle
TAEM
Terminal Area Energy Management
LQR
Linear Quadratic Regulator
PID
Proportional-Integral-Derivative
VSS
Variable Structure System
L/D
Lift Drag Ratio
MPC
Model Predictive Control
ADP
Approximate Dynamic Programming
MPSP
Model Predictive Static Programming
DOF
Degree of Freedom
DI
Dynamic Inversion
I.
Introduction
There is a renewed interest across various space agencies around the world to design economically viable Reusable Launch Vehicles (RLVs) for future space missions to bring down the cost of accessing the
space. Where as the philosophy of repeated launch by the same vehicle sounds interesting and attracting, a
major challenge posed in such missions is that of atmospheric reentry. During this phase, severe constraints
structural load limit and like heat flux constraint come into action, which must be explicitly accounted for and
managed well for the vehicle safety. During this phase the vehicle also needs to fly within upper and lower
bounds of angle of attack to maintain its controllability as well as to manage dissipation of its energy. Moreover, even if the vehicle is unmanned (which is true for most of the next generation RLVs), it is desirable to
fly within the specified bank angle bounds in order not to excite too much the aerodynamic coupling between
longitudinal and lateral dynamics as well as to avoid sharp turnings. In addition to the path constraints, at the
end of the reentry segment the vehicle also needs to meet the desired terminal conditions in terms of desired
position and velocity vector components (which include latitude, longitude, altitude, velocity magnitude,
flight path angle and heading angle) so that the vehicle can be successfully recovered. Moreover, the reentry
3
vehicles are usually unpowered, and the vehicle gets only one chance to land safely. Hence the guidance and
control logic during the reentry must be designed with extreme care so as to have sufficient robustness for
both state perturbations as well as modeling inaccuracies.
In many missions, the ultimate aim is to recover the vehicle in a runway (e.g. like the space shuttle
mission) through an appropriate ‘terminal area energy management’ (TAEM) guidance [1, 2], followed by
appropriate landing logic through typical glideslope and flare. If implemented successfully, this ensures
minimal refurbishing requirement for the vehicle and the turn around time can be short, which can play a
critical role in bringing down the overall cost. However, before initiating the TAEM and automatic landing
logic, it is very crucial to bring the vehicle safely through the reentry corridor to a specified basket in state
space, which is the main aim of this paper. Hence, the problem of TAEM and automatic landing are not
explored here. Moreover, note that the TAEM and automatic landing is not a must in all missions. In simpler
missions, the vehicle is rather required to reach close to a specified final coordinates with sufficiently reduced
velocity from where it can glide to the sea, possibly with the help of a parachute [3]. Such missions are
typically common in initial flight trials to demonstrate the soundness of vehicle design and reentry technology
and/or if the land mass for constructing a runway is not available at a feasible location after the mission of
the launch vehicle is over.
A major breakthrough of the guidance design of reentry vehicles can be attributed to the space shuttle
entry guidance [1]. In this design philosophy, first a reference drag profile is computed in an off-line trajectory
optimization algorithm. This reference trajectory (which is critical for controlled energy dissipation) is then
tracked during the actual flight by incorporating a gain-scheduled PID control design logic. Later it has
been proposed to eliminate the requirement of tedious gain scheduling by substituting it with a dynamic
inversion design [4]. Subsequently, a number of ideas appeared following this basic philosophy of tracking a
reference drag profile using various tracking control design methods. For example, philosophies such as the
gain scheduled LQR design [5, 6] and receding horizon control [7] have been proposed in the literature.
Irrespective of the control design method used, a major drawback of a drag tracking approach, however,
is the over-dependence on the reference profile. For any perturbed flight condition, such a logic forces
the flight vehicle to come back to the reference profile and then keep tracking it. Any reentry guidance
design that typically generates an optimal trajectory offline and then relies on the philosophy of ‘neighboring
4
optimal control’ [8] by forcing to merge the actual trajectory with it is typically not good because of several
reasons. First, these techniques lack the operational flexibility as the onboard trajectory redesign is restricted
to the vicinity of reference profiles computed offline. To bring in operational flexibility (such as choosing a
different runway in case of bad weather), many such reference profiles need to be pre-computed and stored
onboard. Where as unlike earlier days storage space is no more a restriction, one should be careful in
selecting an appropriate reference profile and/or switching between them, which becomes quite a tedious
task. Also, it becomes a vehicle specific logic and hence looses generality. Moreover, in such an approach
the overall guidance that acts on the vehicle in a typical mission is not truly optimal (at the best it can only be
suboptimal). To avoid such drawbacks, the ideal approach would be to carry out the trajectory optimization
process online. Depending on the actual flight condition, ideally a new reference profile itself should be
obtained for making the guidance truly optimal. However, this is in general impossible to carry out online
since dynamic optimization problems are computationally intensive, which are traditionally impossible to
solve in real time using classical techniques and their variants (e.g. using gradient method [9]). Hence,
development of efficient algorithms is a must to solve trajectory optimization problems online.
There have been some attempts in the recent literature to generate feasible reentry trajectories online.
Roenneke [10] has proposed an adaptive entry guidance algorithm based on autonomous onboard trajectory
planning and nonlinear trajectory tracking. This approach essentially eliminates the need for a reference drag
profile and the commanded trajectory to the target is computed by maximizing the vehicle’s range capability.
Cavallo and Ferrara [5] have used a combination of a linear quadratic regulator (LQR) and a variable structure
system (VSS) approach. The LQR design minimizes the deviations from the expected trajectory and the VSS
approach essentially strives to point the velocity vector towards the target thereby minimizing the heading
angle error. Lu [2] presented an algorithm for onboard orbital entry trajectory generation using a quasiequilibrium glide condition to reduce the dimensionality of the problem for meeting inequality trajectory
constraints. The longitudinal and lateral profiles were established through a one-parameter search problem
and bank angle reversals at appropriate points in the trajectory were also found. Shen [11] has provided
a dynamic onboard logic for bank angle maneuvers based on the crossrange profile. Joshi et al. [12] have
employed a predictor corrector approach for the Reusable Launch Vehicle guidance problem where terminal
errors are predicted numerically and then control variables are updated to correct the errors.
5
Nonlinear optimal control theory [8] is the right tool to address a number of challenging trajectory
optimization problems in general, including the reentry guidance. This is because it can naturally handle the
path and terminal constraints, while simultaneously optimizing a meaningful performance index. However,
many of the reentry guidance techniques mentioned above are not based on nonlinear optimal control theory.
This is because the theory, if viewed from a calculus of variations approach, essentially leads to a two-point
boundary value problem and lands up in the issue of ‘curse of complexity’, which in turn requires iterative
solutions leading to the concern of computing time and convergence risk. On the other hand, if viewed from
a dynamic programming formulation, it again leads to the computational bottleneck, known as ‘curse of
dimensionality’ [8]. Recently some fast computational algorithms have been proposed in the literature, but
many of them are not fast enough for aerospace applications, where the available computational time window
is very small (a few milliseconds).
However, combining the philosophies of model predictive control [13] and approximate dynamic programming [14], an innovative computationally efficient technique has been proposed recently to solve a class
of finite horizon optimal control problems with terminal constraints. In addition to its similarity with MPC
and ADP designs, since this new technique is essentially formulated in the framework of static (parametric) optimization, it has been named as “model predictive static programming" (MPSP) [15]. Innovations of
the MPSP technique can be attributed to the following facts: (i) in contrast to typical two-point boundary
value problems in optimal control formulations, it rather demands only a static costate vector (of the same
dimension as the output vector) for the control history update, (ii) the costate vector (and hence the control
history update) has a symbolic solution and (iii) the sensitivity matrices that are necessary for obtaining this
symbolic solution can be computed recursively. Ideas like ‘iteration unfolding’ [16] can also be incorporated
to enhance the computational efficiency further (at the cost of minor compromise on the optimality of the
solution). The technique essentially brings in the philosophy of trajectory optimization into the framework
of guidance design, which in turn results in very effective guidance logic. Recently, the MPSP technique has
been applied to various guidance problems in aerospace engineering with promising results [15, 17, 18].
An alternate promising technique that leads to computationally efficient solutions of optimal control
problems is the ‘pseudospectral method’ [19, 20], which has also been used for reentry guidance problems.
However, it has many tuning issues, including careful selection of basis functions and collocation points
6
(which need to be non-uniform). For successful implementation, it also demands that the user understands
complex mathematical concepts like ‘co-vector mapping principle’. On the other hand, the MPSP technique presented in this paper is rather simpler and straightforward. For example, there is no need of getting
restricted to a non-uniform grid (which brings in additional difficulties for mechanization) and it doe not
demand complex mathematical concepts. Moreover, unlike Pseudospectral methods, MPSP is a method in
itself and does not rely on any numerical techniques for parametric optimization. As a compromise, however,
the technique is still under development and at this moment is not as matured as the Pseudospectral methods.
However, as mentioned above, it is capable enough to address many complex real-life problems [15, 17, 18].
Using the MPSP technique, a suboptimal reentry guidance technique is presented in this paper for a
Reusable Launch Vehicle (RLV). It is worth mentioning that in an earlier related work [17], a reentry guidance problem only in pitch plane was successfully solved by the third author of this paper along with his
other coworkers using the MPSP technique. That problem, however, assumed no bank angle maneuvers, i.e.
bank angle was assumed to be maintained at zero throughout the flight corridor. This was possible as there
was no restriction on the final position of the vehicle after reentry. However, for operational flexibility (i.e.
with better management of vehicle position after reentry) as well as to allow slow dissipation of energy by
giving more flight time, having a strategy for bank angle manipulation (with bank angle reversal) is always
preferable. Note that this brings in an order of magnitude complexity into the problem formulation, which
is successfully addressed in this paper. Another noticeable difference is the ‘specific energy based formulation’ as opposed to a ‘time based formulation’ in the earlier work, [17] which makes the new guidance logic
operate in true feedback sense.
This guidance strategy presented here essentially shapes the trajectory of the RLV by predicting the
necessary angle of attack and bank angle that the vehicle should simultaneously execute. The formulation
uses the nonlinear vehicle dynamics with spherical and rotating earth, hard constraints for desired terminal
conditions and an innovative cost function having several components with associated weighting factors
that can account for path and control constraints in a soft constraint manner, thereby leading to smooth
solutions of the guidance parameters. However, the angle of attack solution comes out of the MPSP guidance
directly. The bank angle command generation is done in two steps. First, the reference heading angle
profile is chosen from the converged solution of the MPSP guidance. Next, using the desired heading angle
7
profile as a command tracking problem, the corresponding bank angle profile is obtained through a dynamic
inversion [21] formulation. Such a two-loop synthesis of bank angle leads to better management of vehicle
trajectory and avoids mathematical complexity as well. Note that the initial guess history for heading angle
in MPSP design is calculated using spherical trigonometry [22] considering earth as a sphere. By choosing
appropriate weights for the heading angle profile update, a good bank reversal strategy is also obtained.
Overall, the bank angle profiles obtained are smooth. Moreover, all bank angle maneuvers have been confined
to the middle of the trajectory and the bank angle is ensured to be near zero at the end of the reentry, which
is very much desirable. The normal load path constraint is minimized by manipulating the angle of attack
profile in those parts of the trajectory where the normal load is close to its boundary. This is achieved by
selecting an appropriate weighting factor (an exponential function of the load profile) in the cost function.
In this paper, the proposed technique has been validated using the nonlinear point mass dynamics of
a realistic reusable launch vehicle with spherical and rotating earth. In addition to nominal case results, it
has also been demonstrated that the proposed guidance has sufficient robustness both for state perturbations
(which may arise from noise input) as well as parametric uncertainties in the model (which can arise from
inaccurate aerodynamic and inertia models). It has been found that the proposed guidance algorithm could
successfully generate feasible trajectories satisfying all constraints. Moreover, the algorithm has been found
to converge very fast with very limited number of iterations. Owing to its computational efficiency and
good robustness, the authors sincerely believe that the MPSP reentry guidance technique presented in this
paper is quite promising in general. Moreover, with the advancement of the computing technology with fast
processors, it holds promise for implementation in onboard processors in the near future.
II.
A Brief Summary of MPSP Design
Even though the MPSP technique has been presented recently in other publications [15, 17, 18], a brief
summary of the salient steps are included in this section for completeness of the paper. One may notice slight
variations of the following algebra in different literature, which may arise due to the selection of different
cost function depending on the necessity of the problem.
To begin with, a discrete (or discretized) form of the system dynamics is considered in the MPSP algorithm. Let X i ∈ ℜn , U i ∈ ℜm , Y i ∈ ℜ p denote the state, control and output variables respectively in the ith
8
iteration, where i = 1, 2, . . . represent the iteration index. Let Xki , k = 1, 2, . . . , N and Uki , k = 1, 2, . . . , N − 1
be the state and control solution respectively at the ith iteration, where X1i represents the given initial condition (which remains same for all i) and Uk1 , k = 1, 2, . . . , N − 1 represents the ‘guess control history’. The
discretized state and output equations in the ith iteration can be represented as
i
Xk+1
= Fk Xki , Uki
Yki = H Xki
(1)
(2)
Like a typical optimal control solution approach, the idea is to predict the system behavior with the most
update control history (starting from an initial guess solution) and then to quickly update it with the error
information available at the final time. Note that like other algorithms, a fairly good guess history is also
recommended to begin the iteration process so that the algorithm converges and converges quickly. However,
the method to obtain a good guess control history is obviously problem specific and for the reentry problem
discussed in this paper, it has been addressed in detail in Section III E. The primary objective is to obtain a
suitable control history Uki , k = 1, 2, . . . , N − 1 at (with as less number of iterations as possible), so that the
output at the final time step YNi goes to a desired value YNd , i.e. YNi → YNd for some i.
The error in the final output at iteration step i is defined as ∆YNi , YNi − YNd . From Eq.(2), taking Taylor
series expansion and introducing small error approximation (thereby neglecting higher order terms) yields
△YNi ≈ dYNi =
∂ YN
dX i
∂ XN (X i ) N
N
(3)
However from Eq.(1), again using the Taylor series expansion and introducing small error approximation,
the error in the state at time step (k + 1) at iteration step i can be expressed as
i
dXk+1
=
∂ Fk
∂ Fk
dXki +
dU i
∂ Xk (X i ,U i )
∂ Uk (X i ,U i ) k
k k
k k
(4)
where dXki is the error in the state and dUki is the error in the control solution at time step k and iteration i.
9
Expanding dXNi as in Eq.(4) (for k = N − 1) and substituting it in Eq.(3) leads to
dYNi
!
∂ YN
∂ FN−1
∂ FN−1
i
i
dX
+
dU
=
∂ XN (X i )
∂ XN−1 (X i ,U i ) N−1
∂ UN−1 (X i ,U i ) N−1
N
N−1 N−1
N−1 N−1
(5)
i
i
Similarly the error in state at time step (N − 1), dXN−1
can be expanded in terms of the errors in state dXN−2
i
and control dUN−2
at time step (N − 2), and so on. Continuing the process until k = 1, one obtains
i
dYNi = A dX1i + B1 dU1i + B2dU2i + . . . + BN−1 dUN−1
∂ YN
∂ FN−1
∂ F1
where A ,
...
∂ XN (X i ) ∂ XN−1 (X i ,U i )
∂ X1 (X i ,U i )
N
N−1 N−1
1 1
∂ YN
∂ FN−1
∂ Fk+1
∂ Fk
Bk ,
...
∂ XN (X i ) ∂ XN−1 (X i ,U i )
∂ Xk+1 (X i ,U i ) ∂ Uk (X i ,U i )
N
N−1 N−1
k+1 k+1
k k
(6)
(7)
where k = 1, . . . , (N − 1). Since the initial condition is specified, there is no error in the first term, which
means dX1i = 0. With this Eq.(6) reduces to
i
dYNi = B1 dU1i + B2dU2i + · · · + BN−1 dUN−1
(8)
It can be pointed out here that the sensitivity matrices Bk , k = 1, . . . , (N − 1) in Eq.(7) can be computed
recursively (see [15] for details), which saves a substantial amount of computational time and makes the
technique very efficient.
Note that Eq. (8) has (N − 1)m unknowns and p equations. Since usually p ≪ (N − 1)m, Eq.(8) is an
under-constrained system of equations. This paves the way for meeting additional objectives. This situation
is exploited by formulating cost functions that can be minimized (or maximized) in addition to satisfying the
constraint in Eq. (8). An assumption is made here that the guess control history is fairly good and close to
optimal. In practice, a problem-dependent wise selection of control guess history should be done (note that a
control guess history specific to the atmospheric reentry problem discussed in this paper has been discussed
with sufficient detail in Subsection III E). With this assumption, the updated control history should remain
10
close to the previous history. Hence, the performance index to be minimized can be chosen as
J=
1 N−1
∑ (dUki )T Rk (dUki )
2 k=1
(9)
where dUki is the corresponding ‘error’ in the control at iteration i that needs to be subtracted from the
previous control value to obtain the new updated control value. Rk > 0 is a time-varying weighting matrix
in general, which needs to be chosen judiciously by the control designer. Note that the performance index
in Eq.(9) is used mainly to demonstrate the MPSP algorithm. The actual cost function with the associated
algebra for reentry guidance problem is included in Section III.
With the above discussion, it is obvious that the cost function in Eq.(9) needs to be minimized subject
to the constraint in Eq.(8). Note that Eq.(8) and Eq.(9) form an appropriate constrained static optimization
problem, which can then be solved in closed form using static optimization theory [8]. Using the very basic
conditions of optimality followed by necessary algebraic manipulations, it then leads to (see [15] for details)
T −1
i
dUki = −R−1
k Bk Aλ dYN
"
where Aλ , −
N−1
T
∑ Bk R−1
k Bk
k=1
(10)
#
(11)
The updated control at time step k = 1, 2, . . . , (N − 1) is given by
Uki+1 = Uki − dUki
(12)
where Uki , k = 1, . . . , (N − 1) is the previous control history solution. The new updated control solution Uki+1 ,
k = 1, . . . , (N − 1) is then used to propagate the system dynamics in Eq.(1) and output dynamics in Eq.(2).
The iterations are carried out until the objectives are met.
Innovations of the MPSP technique can be attributed to the following facts: (i) in contrast to typical
two-point boundary value problems in optimal control formulations, it rather demands only a static costate
vector (that too of the same dimension as the output vector) for the entire control history update (ii) the
11
costate vector (and hence the control history update) has a symbolic solution and (iii) the sensitivity matrices
that are necessary for this solution can be computed recursively. Because of the above facts, this technique
is computationally very efficient, and hence holds promise for online implementation. Ideas like ‘output
convergence’ to terminate the algorithm and ‘iteration unfolding’ [16] (where the control history is updated
only a finite number of times in a particular time step) can also be incorporated to enhance the computational
efficiency further at the cost of a minor compromise about optimality of the solution. As it turns out and have
been demonstrated in a variety of challenging problems [15, 17, 18], the convergence is usually very rapid
and one needs only a few iterations before the control history converges to the optimal control history.
It needs to be emphasized here that even though the basic idea of the MPSP technique is outlined above
for completeness, the actual form of the cost function in a given problem need not conform to Eq.(9). Depending on the application problem, one can choose an appropriate cost function and this may lead to necessary
modifications in the algebra and the final control expression. The reader is referred to Section III H) for a
detail expression of the selected cost function for the reentry problem discussed in this paper and subsequent
sections for the associated algebra.
III. RLV Reentry Guidance Design Using MPSP
The Reusable Launch Vehicle (RLV) considered here is a technology demonstrator to demonstrate various critical technologies, including new guidance and control algorithms, from flight experiments [3]. It does
not have sufficient energy to launch satellites in their orbit. Instead, it is launched for a suborbital mission,
where it is supposed to lift off and travel outside the atmosphere and, after the separation of the booster, is
supposed to reenter through the atmosphere safely so that it can be recovered. During the reentry, the point
where the dynamic pressure builds up to 1.5 kPa has been assumed to be the initial point of reentry. Owing to
its lesser energy because of the suborbital flight, this dynamic pressure is typically experienced at an altitude
of about 51 km with the reentry velocity of approximately 1800 m/s (with flight path angle of approximately
−150). However, the vehicle can actually start its reentry from a ball of initial conditions around these values
in the state space (including position coordinates and heading angle) and guidance scheme should be capable
of guiding the vehicle from any point within this ball. In other words, irrespective of its initial condition
from within this ball, all along the reentry segment the vehicle has to dissipate its associated potential and
12
kinetic energy in a careful manner without violating the path constraints, namely normal load constraint and
the heat flux constraint. At the end of reentry, it has to meet a desired set of final conditions as well. Because
of these constraints the problem is quite challenging. The way it has been successfully solved is discussed in
this section with all necessary mathematical details.
A. Mathematical Model with Spherical and Rotating Earth
It is assumed that the reentry vehicle is unpowered (which is typically true), mass variation is negligible
and the atmosphere is stationary. The only forces acting on the vehicle are gravity and aerodynamic lift and
drag. Under such circumstance, assuming the vehicle to be a point mass, the equations of motion of the
RLV in three-dimensional space over a spherical, rotating earth are described by the following kinematic and
dynamic equations of motion [23]:
ḣ = V sinγ
(13)
D
− g sinγ + Ω2e r cosφ (sinγ cosφ − cosγ sinφ sinψ )
m
1
g
V
L cosσ −
cosγ +
cosγ + 2 Ωe cosφ cosψ
γ̇ =
mV
V
r
Ω2 r cosφ
(cosγ cosφ + sinγ sinφ sinψ )
+ e
V
V cosγ sinψ
φ̇ =
r
V cosγ cosψ
θ̇ =
r cosφ
V̇ = −
ψ̇ =
1 L sinσ
V
−
cosγ cosψ tanφ
m V cosγ
r
+ 2 Ωe (tanγ cosφ sinψ − sinφ ) −
(14)
(15)
(16)
(17)
(18)
Ω2e r
sinφ cosφ cosψ
V cosγ
where the aerodynamic lift L and drag D are given by
1
ρ V 2 CL Sre f
2
1
D = ρ V 2 CD Sre f
2
L=
13
(19)
(20)
The aerodynamic coefficients CL and CD are functions of the angle of attack α and can be written as
CL = CL0 + CLα α
CD = CD0 + CDα α
(21)
Note that the coefficients CL0 , CD0 , CLα and CDα are also functions of angle of attack α and Mach number
M. These are available in the form of look-up tables at various values of angle of attack and Mach number,
which are generated using computational fluid dynamics by a separate team of experts. Linear interpolation
technique is used to compute their values for any given values of the Mach number and angle of attack. In
fact, the lift and drag forces also depend on other additional variables such as control surface deflections,
which are small and hence typically not considered in the guidance design. However, the incremental lift
and drag, as well as the moment components generated due to control surface deflections, are typically
part of the detail six degree-of-freedom model, which is beyond the scope of this paper. Indian Standard
Atmosphere data [24] is used to compute atmospheric properties such as air density and temperature, which
are interpolated at a particular height from a tabulated data. Note that atmosphere density ρ used to compute
the dynamic pressure, where as atmosphere temperature T used to compute the speed of sound, which is in
turn needed to compute the Mach number.
B. Objectives of Guidance Design
The objectives of attaining the terminal conditions and satisfying the path constraints can be done through
appropriate manipulations of the guidance parameters, namely the angle of attack and the bank angle. In
addition, the angle of attack profile is constrained to lie within a minimum and a maximum bound that are
functions of mach number. Similarly, it should not build up high bank angles to avoid problems related to
complicated aerodynamics and difficulty in reversal maneuver. Moreover, the bank angle profile must have
reversals preferably be at the middle part of the trajectory, whereas towards the end of the trajectory the bank
angle should be as close to zero as possible (at the beginning it should also be close to zero, since the vehicle
is expected to enter the atmosphere with near-zero bank angles as well). Note that the path constraints,
namely the normal load constraint and control bounds, constitute a small entry corridor for the vehicle in the
reentry phase through which the vehicle must travel to meet the desired terminal constraints.
14
1.
Path Constraints
Heat Flux Constraint [3, 25]
11030
√
Rn
ρ
ρsl
0.5 Vr
Vco
3.15
≤ Qmax
(22)
Normal Load Constraint [3, 25]
L cosα + D sinα
< Nmax
m
(23)
where the limiting value of the heat flux Qmax is 60 W /cm2 and normal load factor Nmax is 3g. It turns out
that the heat flux constraint is not an active constraint for this mission as it is a suborbital mission (since the
initial energy build up is not as high as an orbital mission, the discipation requirement is not quite high).
In order to maintain sufficient controllability as well as to avoid the stall condition, the angle of attack α
is constrained by the following relationship
αmin (M) ≤ α (M) ≤ αmax (M)
(24)
where αmin (M) and αmax (M) are the minimum and maximum values of the angle of attack, which are functions of Mach number M. Note that the α profile should remain around the middle angle of attack boundary at
a given mach number so that there is enough margin on both sides for the control action in case of unexpected
disturbances as well as to cater for transient effects.
The bank angle history is also desired to be limited by appropriate bounds as dictated by the following
constraint
σmin ≤ σ ≤ σmax
(25)
Note that even if the vehicle is unmanned (which is true for most of the next generation RLVs), it is desirable
to fly within the specified bank angle bounds in order not to excite too much the aerodynamic coupling
between longitudinal and lateral dynamics as well as to avoid sharp turnings. In this paper, the bank angle is
15
constrained to lie within ±450.
2.
Terminal Constraints
The vehicle has to meet the terminal constraints defined in terms of the final height h, final velocity
V and final flight path angle γ at the end of reentry phase. The vehicle must also reach a desired range of
terminal coordinates at the end of the reentry phase.
h f = hd , V f = V d , γ f = γ d , φ f = φ d , θ f = θ d
(26)
where the desired values of final height (hd ), final velocity (V d ), final flight path angle (γ d ), final latitude (φ d )
and final longitude (θ d ) are taken from the reference trajectory that was designed earlier using a gradient
based classical algorithm. Note that even though the formulation has sufficient generality, the final position
constraint is somewhat relaxed here (with larger tolerance bounds) as the task of guiding the vehicle precisely
to the exact final location is that of the terminal area energy management phase, which is not considered here.
3.
Smoothness of Guidance Parameter Profiles
In addition to this, the converged solution of the guidance parameters α and σ that satisfies the aforementioned objectives should preferably be continuous and smooth. This is ensured of by including additional
terms in the cost function that ensure sufficient smoothness and minimum deviation in the updated control
during the update process. Moreover, at the end of reentry, the bank angle profile should ideally take the
form σt=t f = 0 and σ̇t=t f = 0 which ensures that there is sufficient lateral stability at the end of the trajectory.
This makes the tasks of the terminal area energy management phase simpler facilitating a smooth flight from
the end of reentry to the landing or splashdown.
C. System Dynamics with Specific Energy as Independent Variable
The specific energy (i.e. total mechanical energy of the vehicle per unit weight) is chosen as the independent
variable in this guidance design mainly because of two reasons: (i) it eliminates the need to select a final time
(which is a difficult task) and hence also avoids the additional task of optimizing its selection and (ii) it makes
the design operate on a true feedback sense and guidance command generation depends on the current energy
16
and not on the current time. Note that such a change of variable is possible as energy is a continuous and
monotonic variable with respect to time. This is because, due to absence of thrust, only energy dissipation
can take place and hence it can only decrease with time. Theoretically speaking, the kinetic energy of the
vehicle has two components, one due to the body’s center of mass translational kinetic energy and the second
due to the energy of rotation around the center of mass. In this case, the kinetic energy component due to the
rotation of the body around its center of mass is assumed to be negligible compared to the component due to
translational kinetic energy. Therefore the specific energy can be expressed as
e =
V2
+h
2g
(27)
The initial value of specific energy is known from the initial conditions of the vehicle and the final value of
specific energy is defined from the desired terminal velocity and terminal height. Taking time derivative of
Eq. (27) and substituting for V̇ and ḣ, yields the expression for ė. Subsequently, ė has been used to obtain the
set of dynamic equations with energy as the independent variable, such as dh/de = ḣ/ė, dV /de = V̇ /ė etc.
Note that for rest of the paper ‘energy’ means ‘specific energy’.
Note that with this approach, the time t becomes a state variable and t ′ = dt/de is integrated along
with other states to have an idea about the final time needed, which differs with each initial condition. The
problem therefore transforms into a finite energy problem, eliminating the need to predict a final time apriori.
Unlike a finite time based design, it does not unnecessarily put an extra constraint on the problem. One can
also notice that the height ḣ equation defined in Eq. (13) is eliminated in the energy domain equations and
knowing e and V naturally yields the information about h from Eq.(27).
D. Guidance Design Philosophy
One can notice that the longitudinal dynamics is strongly coupled with the angle of attack, the manipulation
of which can cause the error in the terminal velocity and terminal flight path angle to go to the desired values.
Hence, angle of attack is a natural choice as a control variable. Similarly, a proper heading angle profile will
take the vehicle in the direction of the desired terminal position and hence one is tempted to select it as
another control variable. However, a heading angle profile cannot be controlled by manipulating the body
rate equations in the flight control design which is based on the six-DOF dynamics of the vehicle. To avoid
17
this difficulty, even though the heading angle profile is generated from the MPSP guidance, an appropriate
bank angle history is generated to track this heading angle history (a bank angle profile can be controlled
by manipulating the roll rate of the vehicle in the flight control design). Note that care is taken to design
the lateral profile in such a way that the lateral profile is smooth and bank angle maneuvers (including bank
angle reversals) occur in the middle of the trajectory. As evident from the above discussion, the choice of
this design structure has strong physical and mathematical foundation and hence leads to a robust design.
The block diagram of this guidance design philosophy is illustrated in Figure 1. Note that in this subsection
as well as in rest of the paper the variables with no superscript means the superscript i and the variables with
superscript p means the superscript (i − 1) in the context of the MPSP algorithm.
Predictor
State Propagation
Output Error
Convergence
C o n tro l G u ess
U
0
Z k 1
Yk
C o n verg ed
Fk ( Z k , U k )
(\ * \ ) k\ (\ * \ )
YN Y
d YN
h(Z k )
0
*
N
dYN o 0 ?
S o lu tio n
­° D k D k p d D k
Uk ®
p
°̄\ k \ k d\ k
m in d U , m in N Z
Dynamic Inversion
MPSP
Corrector
Fig. 1 Guidance Design Philosophy
E. Selection of Guess Histories of Control Variables in MPSP
After selecting the control variables as the angle of attack and bank angle, like any optimal control
solution approaches, MPSP also needs guess histories of the control variables to start with (which is then
18
rapidly updated online). The middle of the angle of attack bounds is chosen as the guess value for angle of
attack, where as spherical trigonometry is used to determine the nominal heading angle to move towards the
desired location. Details of this selection are discussed next.
1.
Angle of Attack Guess
The angle of attack has upper and lower boundary constraints given by Eq. (24). The operating region
for the angle of attack is therefore constrained to this region for a given mach number. This fact therefore
motivates the use the middle values of the boundary as the α guess history to allow good flexibility during the
control update as the probability of hitting one of the bounds becomes lesser. Note that because of the fact
that a uniform grid point in time does not necessarily lead to same uniform grid points in energy variable, the
cubic spline based interpolation approach is adopted with the available data points to select an appropriate
value. In other words, the available middle values of the angle of attack boundary are considered and a cubic
spline is fitted with these data points. Selection of spline interpolation makes the α guess history smooth as
well.
2.
Heading Angle Guess
Heading angle is primarily used for pointing the vehicle in the direction where it ideally should fly
to reach the target. This objective is achieved through bank angle maneuvers. Since the heading angle is
directly coupled with the latitude-longitude dynamics, the heading angle is considered for manipulation to
attain the desired coordinates. Subsequently, a corresponding bank angle profile that leads to the heading
angle maneuver is generated by using the heading angle dynamics in a dynamic inversion sense. Spherical
trigonometry laws over a spherical earth are used to obtain the nominal heading angle profile [5, 26]. The
desired heading angle depends on the current coordinates (φ , θ ) and the desired final coordinates (φ f , θ f )
coordinates. Figure 2 shows a spherical earth with the start and end of the reentry phase represented by
points A and B respectively. The dashed line between points A and B represents the curvilinear abscissa or
the ground trace of the trajectory followed by the vehicle. Using the law of cosines [22] in the spherical
triangle NACB, the desired heading angle ψ̃ is obtained as a function of φ and θ , details of which are omitted
here for brevity. An interested reader can see the details in [5].
Note that the computations involved in the selection of guess histories for both α and σ are very minimal
19
Fig. 2 Heading Angle Guess using Spherical Geometry
to start the solution approach proposed in this paper.
F. System Dynamics in MPSP Formulation
The state vector considered for the MPSP guidance design is Z , [V γ φ θ ] T . The remaining variables
are considered as energy-varying parameters (like time-varying parameters). The dynamics of h is ignored
since once velocity V is known as a function of energy e, height h automatically gets constrained as per the
relationship e = h + V 2 /(2g). Moreover ψ is considered as a ‘control variable’ in MPSP and its dynamic
equation is kept aside for σ computation. Moreover, even though time t is considered as a state and dt/de
equation is introduced to compute the evolution of time as a function of energy (and hence all variables can
be plotted against the corresponding time), t does not explicitly appear in any other equation and hence the
dt/de equation has been ignored.
The selected state variables are normalized next by defining a set of ‘normalized states’ given by
Zn , [Vn γn φn θn ] T where Vn ,
V
V∗
, γn ,
γ
γ∗
, φn ,
φ
φ∗
, θn ,
θ
θ∗ .
Here V ∗ , γ ∗ , φ ∗ , θ ∗ are
the normalizing values, taken as the corresponding desired terminal values. The control vector is given by
U , [α ψ ] T . Note that the control vector here is not normalised. The normalized system dynamics are
′
represented as Zn , f (Zn , U), where the superscript ′ stands for derivatives with respect to energy e. As the
20
MPSP technique starts with a discretized state equation, using the Euler integration approach the discretized
state equations are written as
Znk+1 = Fk (Znk , Uk ) = Znk + ∆e f (Znk , Uk )
G.
(28)
Output Vector in MPSP Formulation
Since the objective is to drive the error in the final velocity, final flight path angle and final coordinates, the
normalized output vector is chosen as
Yn , [ Vn γn φn θn ] T
(29)
Note that since by definition e f = h f + V f2 /(2g), if V f goes to the desired value at e f , then h f must go to its
desired value as well. Hence there is no need to include it as a component of the output vector.
H.
Cost Function Selection
Selection of a suitable cost function to optimize is one of the key features of any optimal control formulation
and should be done with utmost care. While the terminal constraints can be directly inserted into the MPSP
formulation for optimal guidance design, the path constraints are dealt indirectly through appropriate selection of the cost function, along with associated weighting factors. In addition, continuity and smoothness
in guidance parameters should be maintained as much as possible as these will eventually be tracked by the
inner autopilot loop (which is not within the scope of this paper). Keeping these objectives in mind, the
following cost function was selected to be minimized
J = J1 + J2 + J3
(30)
where,
J1 =
1
2
T
∑N−1
k=1 dUk Rd dUk , for Rd > 0, minimizes the deviation of the updated control from its
previous value. It ensures that the updated control profile remains in the vicinity of the previous
control profile. This is done with the assumption that the initial guess histories of angle of attack and
21
bank angle (which are chosen carefully) are fairly close to the optimal solution.
J2 =
1
2
p
p
T
∑N−1
k=2 (Uk − Uk−1 ) RS (Uk − Uk−1 ) , for RS > 0, is used for additional smoothness of the
updated control profile, where superscript ‘p’ stands for the previous value. Note that smoothness is
achieved primarily when the difference between Uk and Uk−1 is minimized. However, if one chooses
a term (Uk − Uk−1 ) in the cost function, it leads to complicated algebra which should be avoided
as the aim here is to obtain a closed form control update to retain computational efficiency. From
Figure 3, (Uk − Uk−1 ) is given by length CD. However, CD can also be minimized by simultaneously
minimizing AB, AD and BD. As the previous control profile is assumed to be smooth (which is true
for guess histories as well), AB minimization is ensured. Moreover, J1 essentially minimizes segment
BD. Hence, minimization of segment AD by J2 (in conjunction with minimization of J1 ) is sufficient
to ensure smooth control profiles.
p
p
U k-1
Uk
A
B
C
D
Uk-1
Uk
Fig. 3 The geometry of control smoothness.
J3 =
1
2
∑N−1
k=1 ANL e
p
k
BNL NZ
NZk , for ANL , BNL > 0, minimizes the normal load along the path of the
vehicle. Note that the exponential weight is a function of the previous value of the normal load profile.
Although the path constraints specify maintaining the normal load values below Nmax , simulation studies for this problem showed that the normal load lies in the vicinity of Nmax only for a small duration of
the flight. Thus, the exponential weight activates the minimizing function only in specific areas of the
trajectory. It also ensures that the smoothness of the control profile is not compromised at all points on
the trajectory and J1 and J2 dominate the update process when the normal load is significantly lower
than its upper bound.
22
The cost function from Eq. (30) now becomes
1
2
J =
N−1
∑
dUkT Rd dUk +
k=1
1
2
N−1
∑
k=2
p
p
(Uk − Uk−1
)T RS (Uk − Uk−1
) +
1
2
N−1
∑
ANL e
p
k
B NZ
NZk
(31)
k=1
p
Using Uk = Uk − dUk and the terminal constraint given by Eq. (6), the augmented cost function is given by
1
J¯ =
2
N−1
∑
dUkT Rd dUk +
k=1
+
1
2
N−1
∑
k=1
ANL e
p
k
B NZ
1
2
N−1
∑
k=2
T
(32)
Bk dUk )
(33)
p
p
(−dUk + Ukp − Uk−1
) RS (−dUk + Ukp − Uk−1
)
NZk + λ T (dYnN −
N−1
∑
k=1
From the basic condition of optimality concerning the current problem objectives,
∂ J¯
∂ dUk



 




 Rdα 0   d αk   RSα 0   d αk 






+





 




d ψk
d ψk
0 Rdψ
0 RSψ









p

  RSα 0   α    k  − BT
= −
k 2×4 [λ ]4×1




p


ψ
0
R

Sψ




 k 



p


p  ∂ NZk ∂ d αk

 RSα 0   αk−1 
BN


 + ANL e Zk 

 +






p


ψ
∂ N ∂ dψ
0 R
Sψ
∂ J¯
∂ dUk
Zk
k−1




  Rdα 0   d αk







=
0 Rdψ
d ψk






 − BTk
[λ ]4×1
2×4


k
....k > 1




(34)

p  ∂ NZk ∂ d αk 

 + ANL eBNZk 




....k = 1
∂ NZk ∂ d ψk




0 
 RS
 Rdα 0 
, RS ,  α
 and RNL ,
Further algebra is explained using the notations Rd , 




0 Rdψ
0 RSψ
ANL e
p
k
BNZ
.
The expression for the normal load factor
∂ NZk
∂ d αk
becomes a linear function in d αk . This allows the update
process to use angle of attack manipulations to reduce the normal load along the path. The expression for the
23
discretized normal load (Nz ) on expansion yields
Dk
Lk
cosαk +
sin αk
m
m
i
qk Sre f h
(CL0 + CLαk αk ) cosαk + (CD0 + CDαk αk ) sin αk
=
k
k
m
h
qk Sre f
CL0k + CLαk (αkp − d αk ) cos αkp cos(d αk ) + sin αkp sin(d αk )
=
m
i
+ CD0 + CDαk (αkp − d αk ) sin αkp cos(d αk ) − cos αkp sin(d αk )
Nzk ,
k
(35)
This expression is partially differentiated with respect to d αk . Using cos(d αk ) ∼
= 1 and sin(d αk ) ∼
= d αk
since the value of d αk is assumed sufficiently small, Eq. (35) then becomes
∂ NZk
= Tk d αk + Wk
∂ d αk
(36)
where
Tk (αkp ) =
qk Sre f
[ −2CLα sin(αkp ) + 2CDα cos(αkp ) − CL0 cos(αkp ) − CD0 sin(αkp )
m
− CLα αkp cos(αkp ) − CDα αkp sin(αkp ) ]
Wk (αk ) =
p
(37)
qk Sre f
p
p
p
p
[ CL0 sin(αk ) − CD0 cos(αk ) − CLα cos(αk ) − CDα sin(αk )
m
+ CLα αkp sin(αkp ) − CDα αkp cos(αkp ) ]
The normal load is not a function of heading angle, and so
∂ NZk
∂ d ψk
(38)
= 0. Representing
∂ NZk
∂ d ψk
= T2k d ψk +W2k ,
Eq. (34) now becomes
 






p


 d αk 
α  
 d αk 

 + RS 
 − RS  k  − BT


R

d
k 2×4 [λ ]4×1








p


d
d
ψ
ψ
ψ
k
k
∂ J¯
k



 

=

∂ dUk 
p
p
p

0
 αk−1 
  d αk   W1k αk
 T1k αk




+

 + RNL 

 +RS 

 



p
p


W2k ψkp
0
T2k ψk
ψk−1
d ψk
24
 ... k > 1



which yields



 Rk dUk − RS U p + RS U p + RNL Wk − BTk λ = 0 . . . k > 1
k
k−1
∂ J¯
=

∂ dUk

 Rk dUk + RNL Wk − BTk λ = 0



 Rd + RS + RNL Tk . . . k > 1
where Rk ,


 Rd + RNL Tk
... k = 1
From the condition of optimality,
dUk =
∂ J¯
∂ dUk
(39)
... k = 1
= 0. Solving for dUk from Eq. (39),



 R−1 (RS U p − RS U p − RNL Wk + BTk λ ) . . . k > 1
k
k
k−1


T
 R−1
k (− RNL Wk + Bk λ )
Again, from the condition of optimality
∂ J¯
∂λ
(40)
... k = 1
= 0. This gives
N−1
dYnN =
∑
Bk dUk + B1 dU1
(41)
k=2
Substituting dUk from Eq. (40) into Eq. (41), one obtains
dYnN =


p
p

−1
T

∑N−1

k = 2 Bk Rk (RS Uk − RS Uk−1 − RNL Wk + Bk λ )



T
+ B1 R−1
1 (− RNL W1 + B1 λ )






 Bk R−1 (− RNL Wk + BTk λ )
k
... k > 1
(42)
... k = 1
dYnN in Eq. (42) can also be written as
dYnN =



 Aλ λ + b λ 1 − b λ 2 − b λ 3 . . . k > 1


 Aλ λ − b λ 3
25
... k = 1
(43)
where
N−1
Aλ ,
∑
T
Bk R−1
k Bk
∑
p
Bk R−1
k RS Uk
∑
p
Bk R−1
k RS Uk−1
∑
Bk R−1
k RNL Wk
k=1
N−1
bλ 1 ,
k=2
N−1
bλ 2 ,
k=2
N−1
bλ 3 ,
k=1
Hence, λ can be computed from Eq. (43) as
λ =



 A−1 (dYnN − bλ 1 + bλ 2 + bλ 3 ) . . . k > 1
λ


 A−1 (dYnN + bλ 3 )
λ
(44)
... k = 1
Substituting λ into Eq. (40), dUk is obtained as


p
p


R−1

k [RS Uk − RS Uk−1 − RNL Wk +



dUk =
BTk Aλ−1 (dYnN − bλ 1 + bλ 2 + bλ 3 )] . . . k > 1






 R−1 [− RNL Wk + BTk A−1 (dYnN + bλ 3 )] . . . k = 1
k
λ
(45)
Thus, the guidance command can now be updated as
Uk = Ukp − dUk ,
(k = 1, . . . , N − 1)
(46)
Note that in order to update the guidance command, it is necessary to evaluate the Bk matrices from Eq. (7)
∂Y n
in Section II. The terms comprising the Bk matrix are ∂∂ZFnk , ∂∂UFk and ∂ Znk . The values of the state
k
k
k
∂Y nk
∂ Fk
th
variables at the N step are taken as the outputs so that ∂ Zn
, I4 . ∂ Zn and ∂∂UFk are matrices
k
k
k
containing the partial derivatives of Fk with respect to Znk and Uk respectively.
As discussed earlier, the Bk matrices are computed recursively which saves a significant amount of
computational effort. Once again, this feature of the MPSP technique is instrumental in enabling the reentry
guidance scheme to be computationally very efficient.
26
I. Bank Angle Computation
In this subsection, a method to obtain the bank angle (σ ) profile from the updated heading angle (ψ ) is
given. The technique of dynamic inversion (DI) [21] is used to accomplish this task. In the present case,
the real control variable is the bank angle, where as the heading angle is an intermediate control variable.
Hence, using dynamic inversion, the necessary bank angle history is found that will result in the vehicle
having the heading given by the heading angle history, as predicted by the MPSP guidance loop. This is done
by enforcing the following first-order error dynamics
(ψ ∗ ′ − ψ ′ ) + kψ (ψ ∗ − ψ ) = 0
(47)
where, superscript ′ denotes derivative with respect to the energy variable e. The value of ψ ∗ in Eq. (47) is
known from the converged solution of the MPSP formulation.
Introducing quasi-steady approximation, ψ ∗′ is assumed to be zero in each guidance energy interval
window, even though its value gets updated at each grid point. Along with this assumption, substituting the
value of ψ ′ from the system dynamics and carrying out the necessary algebra, one can arrive at
m V cos γ h
V
ė kψ (ψ ∗ − ψ ) +
cos γ cos ψ tan φ
L
r
i
Ω2e r
− 2 Ωe (tan γ cos φ sin ψ − sin φ ) +
sin φ cos φ cos ψ
V cos γ
σ = sin−1
(48)
In addition to obtaining a closed form solution for the bank angle, an additional advantage of using
Eq. (48) is that the sin−1 (.) term will always yield values in the interval [−900, 900 ] based on the sign of its
argument (which largely depends on kψ (ψ ∗ − ψ ), by appropriate selection of kψ ). The design is based on
energy domain, so one can use the information of both initial and final energy values for selecting the time
constant τψ , and hence the kψ = 1/τψ value. The time constant τψ is set in such a manner that the tracking
is neither too aggressive nor very sluggish and the bank angle is always maintained within a bound of ± 450 .
The dynamic inversion technique, along with the heading angle as an intermediate control variable, also
incorporate the means to determine bank reversals. First, note that it is desirable to have bank reversals in
the middle segment of the trajectory. This is because in the beginning the vehicle is not expected to have
large dynamic pressure and hence it is difficult to do large bank angle maneuvers (it is not advisable to
27
activate the reaction control system for this). More important, at the end of the reentry the vehicle should
preferably be in the no roll attitude to facilitate good terminal area energy management. Hence, the bank
angle should be close to zero at the end of the reentry phase. In practice, this is done by setting the weight
for d ψ minimization in the control update process large at the start and end of the trajectory and small in
the middle of the trajectory. The weight chosen for d ψ , Rd ψ is illustrated in Figure 4. A hyperbolic tangent
function represented in Eq. (49) is used to generate this weight. Clearly, d ψ will have little room to vary at
the start and near the end of the trajectory, but will have good flexibility in the middle part.
500
450
400
350
R
dψ
300
250
200
150
100
50
0
0
100
200
300
400
500
Energy Steps
600
700
800
Fig. 4 Weighting function for d ψ Minimization
Rd ψk


m k − N6



A − B tanh 

N


6











=
A − B tanh m












5N − k

m

6



A − B tanh


N
k = 1, . . . , N/3
k = N/3, . . . , 2N/3
(49)
k = 2N/3, . . . , N
6
where A and B are constants that define the maximum and minimum value of the weight and the factor m
determines how fast the transition from the maximum to minimum values occurs.
28
IV. Numerical Results
The reentry guidance technique presented in this paper is simulated using realistic vehicle data as generated from the computational fluid dynamics analysis. The mass of the vehicle is assumed constant for the
entire flight since (i) there is no thrust in the vehicles (ii) heat flux being not high ablative cooling mechanism
is not there in the vehicle and (iii) reaction control jet fuel depletion is very small compared to the mass of
the vehicle and can safely be neglected especially in the guidance computation. The nominal initial condition
for reentry as well as the final desired conditions after the reentry phase are tabulated below.
Table 1 Initial Nominal Conditions for Reentry
Variables
Height (h)
Velocity (V )
Flight path angle (γ )
Latitude angle (φ )
Longitude angle (θ )
Value
51000 m
1796 m/s
-15.32 deg
16.0 deg
84.0 deg
Table 2 Final Desired Conditions for Reentry
Variables
Height (h)
Velocity (V )
Flight path angle (γ )
Latitude angle (φ )
Longitude angle (θ )
Value
20042 m
556.59 m/s
- 20.44 deg
14.03 deg
85.93 deg
Note that whereas the desired final condition remains the same, the initial condition can vary and results
are included for a number of initial conditions around these nominal values. Moreover, perturbation studies
with respect to aerodynamic data perturbation is also carried out. Details of this study as well as the results
obtained are included in this section.
A. Selection of Energy Step Size
Since energy is considered as the independent variable in the formulation presented in this paper, its step
size for the state propagation must be chosen wisely. In fact, guidance cycle update for stable vehicles such
as the RLV under consideration typically takes place at 100 ms interval in time domain. Hence, there are
two options (i) to select a variable energy interval that will correspond to 100 ms interval in time domain or
(ii) to select a constant energy interval which will be representative of this situation. Note that option (i) is
difficult to mechanize since without closing in the guidance and control loops a variable energy step size to
represent a constant step size in time is impossible to obtain as it depends on the actual trajectory followed
(which is unknown before designing the guidance loop). Hence option (ii) was selected to be followed in
this paper. With this in mind, open loop simulation were carried out to determine the largest value of the
energy step that has a corresponding time step approximately equal to 100 ms. Note that because energy
29
is a continuous and monotonic function of time (see Fig. 8 for representative plots of energy vs time), such
a mechanization will ensure that the actual time interval remains equal to or higher than 100 ms in reality
and hence computational time delay will not be much of a concern from time delay margin point of view.
On the other hand, since energy is a slowly changing variable (and hence approximately a linear function of
time), the maximum time interval is not substantially higher than 100 ms and hence guidance objective is not
compromised either. From a few open loop simulations, ∆e was finally fixed at −248 m.
B. Convergence condition in MPSP
The convergence bound for terminating the MPSP interation is defined such that the algorithm recognizes whether the solution is good enough or not. Even though a number of possibilities exists for defining a
good convergence condition, it is assumed here that the convergence of the algorithm takes place when all of
the following conditions hold: (i) the absolute error in altitude is less than 25m, (ii) absolute error in velocity
is less than 0.5m/s, (iii) absolute error in flight path angle is less than 0.20 , and (iv) absolute error of the
position coordinates, i.e. latitude and longitude, are individually less than 0.10 . One can clearly notice that
these are quite small, and hence, the terminal accuracy is enforced to be quite good. Moreover, it was also
decided that the solution is ‘acceptable’ only if all the path constraints are also met.
C. Results with Nominal and Perturbations in Initial Conditions
The proposed guidance technique is tested by assuming the nominal as well as perturbed initial conditions at the reentry point. The initial conditions along with the errors in the outputs at the end of the reentry
are given in Table 3, where ‘Base case’ stands for the ‘Nominal case’ and other cases stand for perturbations
around it. From the results obtained, it can be inferred that the algorithm successfully works not only for the
nominal case, but also for a number of cases around it and results in very good final accuracy in each case.
Various trajectory results are plotted in Figs. 5–16. The altitude and velocity trajectories are plotted in
Figs. 5 and 6 respectively. It can be observed that whereas the velocity trajectory in monotonically decreasing,
the altitude trajectory is not so. In fact, after some time, the vehicle actually climbs a bit before coming down
again, thereby allowing itself to get more time to dissipate its energy. This is also the phase of higher lift,
as evident from the normal load plots in Fig. 7. The plots in Fig. 7 also show that the normal load acting
on the vehicle always remain less than 3g for all time in all cases, which is a strong requirement from the
30
Table 3 Different initial conditions and corresponding errors in the outputs at the end of reentry
Case
h (km)
V (m/s)
γ (deg)
Base Case
Case 1
Case 2
Case 3
Case 4
Case 5
Case 6
Case 7
51.000
46.920 (-8%)
52.530 (+3%)
49.980 (-2%)
53.550 (+5%)
47.940 (-6%)
49.980 (-2%))
47.940 (-6%)
1,796
1,867 (+4%)
1,760 (-2%)
1,832 (+2%)
1,743 (-3%)
1,832 (+2%)
1,760 (-2%)
1,849 (+3%)
-15.32
-15.63 (+2%)
-14.86 (-3%)
-15.63 (+2%)
-14.4 (-6%)
-14.55 (-5%)
-14.4 (-6%)
-15.01 (-2%)
Eh (km) EV (m/s) Eγ (deg) Eφ (deg) Eθ (deg)
|hN − hd | |VN − V d | |γN − γ d | |φN − φ d | |θN − θ d |
0.003
0.005
0.008
0.002
0.005
0.013
0.005
0.015
0.059
0.086
0.159
0.033
0.071
0.223
0.093
0.265
0.1656
0.0568
0.002
0.150
0.0474
0.0262
0.1616
0.0515
0.0562
0.019
0.0754
0.0272
0.0703
0.0135
0.0915
0.0058
0.0540
0.018
0.0726
0.0260
0.0679
0.0128
0.0883
0.0053
vehicle safety consideration. The associated profiles for specific energy (which is a function of altitude and
velocity) are shown in Fig. 8, which conforms to the assumption that it remains monotonic throughout the
reentry phase owing to the presence of drag and absence of thrust.
The flight path angle and heading angle trajectories are plotted in Fig. 9 and Fig. 10 respectively. From
Fig. 9 one can observe how the velocity vector keeps changing its direction on its flight path. Moreover,
the change is quite smooth. This essentially due to the smoothness enforcement on α in the cost function
formulation, because of which the α trajectory turns out to be smooth. From Fig. 10 it is clear that the
initial heading angle guess as done through the spherical trigonometry algebra is quite good and only a small
variation about that is necessary to reach the final destination with good accuracy. The variation along the
way is not much either. Moreover, all the variations happen in the middle of the trajectory, which is very
nice. Note that both ψ and ψ̇ stabilize towards the end of the trajectory, thereby driving the bank angle to
zero in all cases.
The latitude and longitude trajectories are plotted in Fig. 11 and Fig. 12 respectively. Here the change is
not as apparent since they all start with the same latitude and longitude (variation of that is studied separately).
However, it should be noted that in each case they end up with ±0.10 error at the end of the reentry (the
desired coordinates at the end of the reentry are 14.030 and 85.930 respectively).
The dynamic pressure and heat flux trajectories are plotted in Fig. 13 and Fig. 14 respectively. First,
Fig. 14 shows that the heat flux is well within the allowable bound (which is put as 40W /cm2 in this paper),
and hence there is no danger to the vehicle. Moreover, the heat flux stabilizes at a very small value of
31
2W /cm2 at the end in all cases. This corroborates the fact that the reentry is actually over at this point of
time. A high dynamic pressure with a low normal load at the end of the reentry also indicate a good control
authority as angle of attack α can then be varied freely after the end of the reentry.
Finally the guidance parameter trajectories (i.e. trajectories of α and σ ) are plotted in Fig. 15 and Fig. 16
respectively. Figure 15 shows that the angle of attack (α ) trajectories are smooth for all cases and remain
fairly close to the middle of the upper and lower boundaries defined by αmax (M) and αmin (M) respectively,
leaving quite a bit of gap on both sides. This is very good as it gives margins for the autopilot to correct
unexpected disturbances on the way. Figure 16 also shows that the actual values are quite small and are well
within the bounds of ±450. However, at this point it makes sense to exercise a bit of caution while inferring
this, since all of these simulation cases start with the same initial latitude and longitude coordinates. Once
that is perturbed, the required bank angles turn out to be much more, as evident from the results presented in
a subsequent subsection.
2000
Base Case
Case−1
Case−2
Case−3
Case−4
Case−5
Case−6
Case−7
Altitude(km)
45
40
Base Case
Case−1
Case−2
Case−3
Case−4
Case−5
Case−6
Case−7
1500
Velocity(m/s)
50
35
1000
30
25
20
200
250
300
350
400
Time (sec)
450
500
550
Fig. 5 Altitude trajectories for initial conditions in
Table 3
500
200
250
300
350
400
Time (sec)
450
500
550
Fig. 6 Velocity trajectories for initial conditions in
Table 3
D. Simulation study with large number of random perturbations
Next, in order to adequately demonstrate the robustness of the proposed technique to variations in the
initial conditions, a large number of simulation studies (100 to be exact) were carried out with random
perturbations in the initial velocity, initial height and initial flight path angle (which turn out to be sensitive
parameters). Table 4 gives the range of these perturbations.
The results obtained from this study demonstrate that terminal and path constraints have been met with
32
5
2.5
3
Base Case
Case−1
Case−2
Case−3
Case−4
Case−5
Case−6
Case−7
2
Base Case
Case−1
Case−2
Case−3
Case−4
Case−5
Case−6
Case−7
2
Energy (m)
Normal Load (g)
2.5
x 10
1.5
1.5
1
1
0.5
0.5
0
200
250
300
350
400
Time (sec)
450
500
0
200
550
Fig. 7 Normal load trajectories for initial conditions
in Table 3
250
300
350
400
Time (sec)
450
500
550
Fig. 8 Specific energy trajectories for initial conditions in Table 3
320
Base Case
Case−1
Case−2
Case−3
Case−4
Case−5
Case−6
Case−7
Flight path angle, γ (deg)
5
0
319
Base Case
Case−1
Case−2
Case−3
Case−4
Case−5
Case−6
Case−7
318
Heading Angle, ψ(deg)
10
−5
−10
317
316
315
314
313
312
−15
311
−20
200
250
300
350
400
Time (sec)
450
500
310
200
550
250
300
350
400
Time (sec)
450
500
550
Fig. 10 Heading angle trajectories for initial conditions in Table 3
Fig. 9 Flight path angle trajectories for initial conditions in Table 3
86
Base Case
Case−1
Case−2
Case−3
Case−4
Case−5
Case−6
Case−7
15.8
15.6
Latitude, φ (deg)
15.4
15.2
85.8
85.6
85.4
Longitude, θ(deg)
16
15
14.8
85.2
85
84.6
14.6
14.4
84.4
14.2
84.2
84
14
200
Base Case
Case−1
Case−2
Case−3
Case−4
Case−5
Case−6
Case−7
84.8
250
300
350
400
Time (sec)
450
500
550
Fig. 11 Latitude trajectories for initial conditions in
Table 3
200
250
300
350
400
Time (sec)
450
500
550
Fig. 12 Longitude trajectories for initial conditions
in Table 3
good accuracy as well. Moreover, the convergence rate of the algorithm is 100% and the number of iterations
33
15
Base Case
Case−1
Case−2
Case−3
Case−4
Case−5
Case−6
Case−7
16
12
10
2
Heat Flux(W/cm )
Dynamic prssure(kpa)
14
Base Case
Case−1
Case−2
Case−3
Case−4
Case−5
Case−6
Case−7
5
0
200
250
300
350
400
Time (sec)
450
500
10
8
6
4
2
550
Fig. 13 Dynamic pressure trajectories for initial conditions in Table 3
200
250
300
350
400
Time (sec)
450
550
Fig. 14 Heat flux trajectories for initial conditions in
Table 3
45
20
40
Base Case
Case−1
Case−2
Case−3
Case−4
Case−5
Case−6
Case−7
15
35
10
30
Bank Angle, σ (deg)
Angle of Attack, α (deg)
500
25
20
15
5
0
−5
10
−10
5
−15
0
−5
200
250
300
350
400
Time (sec)
450
500
550
Fig. 15 Angle of attack trajectories for initial conditions in Table 3
−20
200
250
300
350
400
Time (sec)
450
500
550
Fig. 16 Bank angle trajectories for initial conditions
in Table 3
Table 4 Range of Random Perturbations in Initial Conditions
Variable
Height (h), km
Velocity (V ), m/s
Flight Path Angle (γ ), deg
Base Initial Value Range of Perturbations
51.0
1796
-15.32
± 3.0
± 100
±2
for convergence is found to be generally two (with upper limit being four and lower limit being as small as
one). The normal load constraint is maintained below 3g for each simulation. Detail plots resulting out of
this simulation study are omitted here to contain the length of the paper.
34
E. Simulation Study with Perturbation in Reentry Coordinates
In this part of the simulation study, perturbations are introduced in in the expected initial position of
reentry. Latitude φ and Longitude θ are varied by ±1deg and altitude is varied by ±2.5km. Once again,
even though a large number of simulation were carried out, eight cases were randomly selected for including
representative results in this paper. The numerical values of the actual reentry position as well as the errors
obtained for the final desired position for these eight cases are given in Table 5. Note that the desired
coordinates at the end of the reentry are h(t f ) = 20.0km, φ (t f ) = 14.030 and θ (t f ) = 85.930 respectively and
the actual values obtained are very close to these values.
Table 5 Initial Perturbation in Reentry Coordinates
Case
Base Case
Case 1
Case 2
Case 3
Case 4
Case 5
Case 6
Case 7
Initial
Initial
Initial
Eh (km) Eφ (deg) Eθ (deg)
Altitude h(km) Latitude (φ , deg) Longitude (θ , deg) |hN − hd | |φN − φ d | |θN − θ d |
51.000
49.980
53.550
52.530
48.450
52.530
53.550
49.980
16
16.3
15.4
16.5
15.3
15.7
16.2
15.5
84
84.5
83.5
84.7
83.6
83.7
84.3
83.6
0.003
0.013
0.010
0.006
0.010
0.009
0.005
0.009
0.0561
0.0386
0.024
0.0789
0.0032
0.0384
0.0061
0.0396
0.0542
0.0239
0.0417
0.0388
0.0060
0.0504
0.0047
0.0617
Only a few selected plots are included form this exercise to contain the length of the paper. Figures 17
and 18 represent the altitude and heading angle trajectories. The nature of the altitude variation is as expected
and it is nice to see that the evolution of the altitude variation happens to be quite close to each other in all
cases. Even though the variation of the heading angle is not much in any individual case, the value itself
depends on the initial position of the vehicle, which is once again conforms to the intuition. The separation
between the lowest and highest value happens to be as high as 350 , which was necessary to guide the vehicle
in the desired position direction (which remains the same irrespective of where the reentry starts). The
corresponding latitude and longitude trajectories are shown in Figs. 19 and 20 respectively. It is quite evident
form these plots how they evolve towards the desired value of φ (t f ) = 14.030 and θ (t f ) = 85.930 respectively.
Note that the vehicle manages to reach the vicinity of the desired location within a ±0.1deg accuracy in both
latitude and longitude (which is also quite clearly evident in Table 5).
35
55
Base Case
Case−1
Case−2
Case−3
Case−4
Case−5
Case−6
Case−7
Altitude(km)
45
40
Base Case
Case−1
Case−2
Case−3
Case−4
Case−5
Case−6
Case−7
330
325
Heading Angle, ψ(deg)
50
35
30
320
315
310
305
25
300
20
295
15
200
250
300
350
400
Time (sec)
450
500
550
Fig. 17 Altitude trajectories for initial positions in
Table 5
200
300
350
400
Time (sec)
450
550
86
Base Case
Case−1
Case−2
Case−3
Case−4
Case−5
Case−6
Case−7
15.5
85.5
Longitude, θ(deg)
16
15
Base Case
Case−1
Case−2
Case−3
Case−4
Case−5
Case−6
Case−7
85
84.5
84
14.5
14
200
500
Fig. 18 Heading angle trajectories for initial positions in Table 5
16.5
Latitude, φ (deg)
250
83.5
250
300
350
400
Time (sec)
450
500
550
Fig. 19 Latitude trajectories for initial positions in
Table 5
83
200
250
300
350
400
Time (sec)
450
500
550
Fig. 20 Longitude trajectories for initial positions in
Table 5
The associated guidance parameters, namely the angle of attack α and bank angle σ plots are shown in
Figs. 21 and 22 respectively. It is evident that variation in α is not much as it strongly depends on dynamic
parameters such as velocity magnitude and flight path angle (which have been kept constant in this exercise).
However, the variation of σ is quite a bit here as that plays a role in position correction and hence depends
strongly on where the vehicle starts its reentry. Note that in one case (Case - 2), at one point the value of σ
is relatively high, the peak value being −30.50 (which is still very much within the ±450 bound). This is
because of the fact that in that case the initial condition of the longitude is quite far away from the nominal
case and hence the vehicle needs to be steered away towards the desired location with a bit of aggressive
maneuver. However, it is once again nice to see that the aggressive maneuver takes place in the middle part
of the trajectory, which is because of the way the cost function is designed. In each case, the vehicle again
36
stabilizes with zero bank angle at the end of the reentry.
45
30
30
20
Bank Angle, σ (deg)
Angle of Attack, α (deg)
35
25
20
15
10
0
−10
10
−20
5
−30
0
−5
200
Base Case
Case−1
Case−2
Case−3
Case−4
Case−5
Case−6
Case−7
40
40
−40
250
300
350
400
Time (sec)
450
500
550
Fig. 21 Angle of attack trajectories for initial positions in Table 5
250
300
350
400
Time (sec)
450
500
550
Fig. 22 Bank angle trajectories for initial positions in
Table 5
F. Simulation study with Perturbation in Aerodynamic Parameters
This section provides simulation results to test the capability of the proposed guidance technique to
recompute the trajectory online, in case of perturbations in the aerodynamic parameters, namely CL and
CD , which strongly effect the dynamics. In the simulation experiment, a ± 10% random variations in the
aerodynamic parameters CL0 , CD0 , CLα and CDα was considered to evaluate the performance of the proposed
guidance logic. An assumption made here is that the percentage error of the coefficient values remain same
throughout the flight trajectory. However, there was a need to excite the guidance logic frequently along the
flight path due to the fact that the actual state was not following the predicted state because of the modelling
discrepancy. In total, 100 trajectories were generated in this exercise by randomly selecting a percentage
error (within ± 10% of the base value) for each aerodynamic coefficient in each simulation.
The outcome of this simulation study indicates (i) all trajectories are maintained relatively smooth for the
full duration of reentry, (ii) the normal load is always maintained below its upper limit of 3g (iii) the angle of
attack trajectories remain close to the middle of the upper and lower bounds, leaving aside sufficient margin
for the autopilot to handle unwanted disturbances, (iv) bank angles are bounded between ±450 and bank
angle reversals happen at the middle of the trajectory. Moreover, it turns out that the algorithm converges
within a few iterations (less than five iterations). These results demonstrate the robustness of the guidance
technique with respect to perturbations in aerodynamic parameters as well as the rapid convergence of the
37
MPSP algorithm. Detail plots and table of results are omitted to contain the length of the paper.
V. Conclusions
Taking the help of recently-developed model predictive static programming, a suboptimal reentry guidance logic is presented in this paper for a reusable launch vehicle in a technology demonstration mission.
This guidance essentially shapes the trajectory of the vehicle by predicting the necessary angle of attack
and bank angle that the vehicle should execute, while satisfying a set of path and terminal constraints. The
guidance law is primarily based on nonlinear optimal control theory and hence imbeds effective trajectory
optimization concepts into the guidance law. In addition to the promising results for the nominal case, it has
also been demonstrated that this guidance also has sufficient robustness for both state perturbations as well
as parametric uncertainties in the model.
It can be mentioned here that the guidance commands generated in this paper has also been realized in
an inner-loop autopilot design using the full six degree-of-freedom model of the vehicle. While designing the
autopilot, extensive attention has been given both for aerodynamic and reaction jet control system designs
as well as their fusion. However, the authors believe that the details of the autopilot design, which includes
a nominal loop as well as an adaptive loop for enhanced robustness for modeling inaccuracies, merits to be
reported in a separate paper.
Acknowledgement
The authors would like to acknowledge the contribution of S. Mathavaraj during the initial phase of
this work through technical discussions. He has also contributed significantly in designing a nonlinear and
robust inner-loop autopilot accounting for the full six degree-of-freedom vehicle dynamics for realizing the
guidance commands.
38
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